Average Error: 53.4 → 0.2
Time: 10.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.995295059253408442856425608624704182148:\\ \;\;\;\;\log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)}\right) + \log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)}\right)\\ \mathbf{elif}\;x \le 0.8946210124348296099938693259900901466608:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.995295059253408442856425608624704182148:\\
\;\;\;\;\log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)}\right) + \log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)}\right)\\

\mathbf{elif}\;x \le 0.8946210124348296099938693259900901466608:\\
\;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right) + x\right)\\

\end{array}
double f(double x) {
        double r164810 = x;
        double r164811 = r164810 * r164810;
        double r164812 = 1.0;
        double r164813 = r164811 + r164812;
        double r164814 = sqrt(r164813);
        double r164815 = r164810 + r164814;
        double r164816 = log(r164815);
        return r164816;
}
↓
double f(double x) {
        double r164817 = x;
        double r164818 = -0.9952950592534084;
        bool r164819 = r164817 <= r164818;
        double r164820 = 0.125;
        double r164821 = 3.0;
        double r164822 = pow(r164817, r164821);
        double r164823 = r164820 / r164822;
        double r164824 = 0.0625;
        double r164825 = 5.0;
        double r164826 = pow(r164817, r164825);
        double r164827 = r164824 / r164826;
        double r164828 = 0.5;
        double r164829 = r164828 / r164817;
        double r164830 = r164827 + r164829;
        double r164831 = r164823 - r164830;
        double r164832 = sqrt(r164831);
        double r164833 = log(r164832);
        double r164834 = r164833 + r164833;
        double r164835 = 0.8946210124348296;
        bool r164836 = r164817 <= r164835;
        double r164837 = 1.0;
        double r164838 = sqrt(r164837);
        double r164839 = pow(r164838, r164821);
        double r164840 = r164822 / r164839;
        double r164841 = -0.16666666666666666;
        double r164842 = log(r164838);
        double r164843 = r164817 / r164838;
        double r164844 = r164842 + r164843;
        double r164845 = fma(r164840, r164841, r164844);
        double r164846 = r164829 - r164823;
        double r164847 = r164846 + r164817;
        double r164848 = r164847 + r164817;
        double r164849 = log(r164848);
        double r164850 = r164836 ? r164845 : r164849;
        double r164851 = r164819 ? r164834 : r164850;
        return r164851;
}

Error

Bits error versus x

Target

Original53.4
Target45.4
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.9952950592534084

    1. Initial program 63.0

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Simplified63.0

      \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + x\right)}\]

    3. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)}\]

    4. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)}\]
    5. Using strategy rm

    6. Applied add-sqr-sqrt0.2

      \[\leadsto \log \color{blue}{\left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)} \cdot \sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)}\right)}\]

    7. Applied log-prod0.2

      \[\leadsto \color{blue}{\log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)}\right) + \log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)}\right)}\]

    if -0.9952950592534084 < x < 0.8946210124348296

    1. Initial program 58.9

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Simplified58.9

      \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + x\right)}\]

    3. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}}\]

    4. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)}\]

    if 0.8946210124348296 < x

    1. Initial program 31.7

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Simplified31.7

      \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + x\right)}\]

    3. Taylor expanded around inf 0.2

      \[\leadsto \log \left(\color{blue}{\left(\left(x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)} + x\right)\]

    4. Simplified0.2

      \[\leadsto \log \left(\color{blue}{\left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right)} + x\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.995295059253408442856425608624704182148:\\ \;\;\;\;\log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)}\right) + \log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)}\right)\\ \mathbf{elif}\;x \le 0.8946210124348296099938693259900901466608:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))